How may unicode-math access the following common power set symbol?

Question

Here is an MWE to obtain a calligraphic P (which is not supported by texgyrepagella, under Unicode-math) with the extra command \setmathfont[range=\symcal]{Asana Math}:

\documentclass{article}

\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{unicode-math}

\setmathfont{texgyrepagella-math.otf}
\setmathfont[range=\symcal]{Asana Math}

\begin{document}
    $\mathcal{P}$
 
\end{document}

I want that my unicode-math documents have access to the calligraphic mathcal P as in this Proof Wiki, for that is the standard power set symbol.

How may it be accomplished?

Answer 1

First off, there is no “standard” power set symbol. The ProofWiki page you cite says at most what some symbols should be used for in ProofWiki. They seem to fancy Knuth's CM Symbols shapes.

For instance, they say 𝒪 is used for big-O notation, but this is generally not true and most of the sources I know use simply O. Whether you use ℬ for the basis of a topology or for whatever else is up to the author/publisher to decide. And, personally, I've never used ℛ for a relation.

You find the power set denoted by 𝒫, but also by a fancy calligraphic variant, or simply by P. Just make it clear at the outset what notation you are using.

This said, I know of no Unicode math font that decided to borrow Knuth's calligraphic, so I can propose a simple solution, namely to load the font yourself. I anyway concur that the calligraphic alphabet offered by TeX Gyre Pagella is unsightly.

\documentclass{article}

\usepackage{amsmath}
%\usepackage{amsfonts}% not with unicode-math
\usepackage{unicode-math}

\DeclareMathAlphabet{\CMcal}{OMS}{cmsy}{m}{n}
\SetMathAlphabet{\CMcal}{bold}{OMS}{cmsy}{b}{n}
\DeclareMathAlphabet{\CMbfcal}{OMS}{cmsy}{b}{n}

\setmainfont{TeX Gyre Pagella}
\setmathfont{texgyrepagella-math.otf}
\setmathfont{NewCMMath-Book.otf}[range={cal,bfcal}]
\setmathfont{texgyrepagella-math.otf}[range=]

\newcommand{\pow}{\CMcal{P}}

\begin{document}

$\mathcal{P}$ $\CMcal{P}$

$\pow(X)$
 
\end{document}

Do use a command, so you're free to change the associated symbol at any time.

You can get as many shapes of the calligraphic P as are available.

\documentclass{article}

\usepackage{amsmath}
\usepackage{unicode-math}

% for Computer Modern calligraphic
\DeclareMathAlphabet{\CMcal}{OMS}{cmsy}{m}{n}
\SetMathAlphabet{\CMcal}{bold}{OMS}{cmsy}{b}{n}
\DeclareMathAlphabet{\CMbfcal}{OMS}{cmsy}{b}{n}

\setmainfont{TeX Gyre Pagella}

% the main math font
\setmathfont{texgyrepagella-math.otf}
% a decent mathcal font
\setmathfont{NewCMMath-Book.otf}[range={cal,bfcal}]
% restore the font dimensions
\setmathfont{texgyrepagella-math.otf}[range=]

\newfontfamily{\aaaa}{Latin Modern Math}[Scale=MatchUppercase]
\newfontfamily{\bbbb}{STIX Two Math}[Scale=MatchUppercase]
\newfontfamily{\cccc}{TeX Gyre DejaVu Math}[Scale=MatchUppercase]
\newfontfamily{\dddd}{Concrete Math}[Scale=MatchUppercase]

\newcommand{\pow}{\CMcal{P}}

% for testing
\newcommand{\fancyp}[1]{\text{#1\symbol{"1D4AB}}}

\begin{document}

$\mathcal{P}$ $\CMcal{P}$

$\pow(X)$

$\fancyp{\aaaa}+\fancyp{\bbbb}+\fancyp{\cccc}+\fancyp{\dddd}$
 
\end{document}

Answer 2

Choose the font whose script alphabet you like the best, and load that as \mathcal. Concrete Math contains one that’s close to the reference you gave:

\documentclass{article}

\usepackage{unicode-math}

\setmathfont{texgyrepagella-math.otf}
\setmathfont{Concrete Math}[range={cal, bfcal}, Scale=MatchUppercase]

\begin{document}
    $\mathcal{P}$
 
\end{document}

But Euler Script (by Hermann Zapf, who also designed Palatino, of which both Pagella and Asana are clones) is another classic one. You can get it from Euler Math.

Answer 3

But TeX Gyre Pagella does provide a calligraphic P. You can use any calligraphic P to denote a power set.

\documentclass{article}

\usepackage{pagella-otf}

\begin{document}
    We denote the power set of \(X\) by \(\mathcal{P}(X)\).
\end{document}